Session III
Please note: The second talk by Fracis Bach was jointly organized with the One World Optimization Seminar.
| Speaker | Title tap/hover for abstract | Materials |
|---|---|---|
| Lars RuthottoEmory University, US | Machine Learning meets Optimal Transport: Old solutions for new problems and vice versa Machine Learning meets Optimal Transport: Old solutions for new problems and vice versa This talk presents new connections between optimal transport (OT), which has been a critical problem in applied mathematics for centuries, and machine learning (ML), which has been receiving enormous attention in the past decades. In recent years, OT and ML have become increasingly intertwined. This talk contributes to this booming intersection by providing efficient and scalable computational methods for OT and ML. The first part of the talk shows how neural networks can be used to efficiently approximate the optimal transport map between two densities in high dimensions. To avoid the curse-of-dimensionality, we combine Lagrangian and Eulerian viewpoints and employ neural networks to solve the underlying Hamilton-Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present numerical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting. The second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators, which are critical in machine learning. We consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward problem relies on manually derived gradients and Laplacian of the neural network and uses automatic differentiation in the optimization. In common benchmark challenges, our method outperforms state-of-the-art CNF approaches by reducing the network size by 8x, accelerate the training by 10x- 40x and allow 30x-50x faster inference. |
video slides |
| Francis BachUniversité PSL, FR joint work with:Lénaïc Chizat |
On the convergence of gradient descent for wide two-layer neural networks On the convergence of gradient descent for wide two-layer neural networks Many supervised learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters, this often leads to convex problems for which many guarantees exist. Models which are non-linear in their parameters such as neural networks lead to non-convex optimization problems for which guarantees are harder to obtain. In this talk, I will consider two-layer neural networks with homogeneous activation functions where the number of hidden neurons tends to infinity, and show how qualitative convergence guarantees may be derived. I will also highlight open problems related to the quantitative behavior of gradient descent for such models. |
video slides |
Video Recordings
Lars Ruthotto: Machine Learning meets Optimal Transport: Old solutions for new problems and vice versa
Abstract: This talk presents new connections between optimal transport (OT), which has been a critical problem in applied mathematics for centuries, and machine learning (ML), which has been receiving enormous attention in the past decades. In recent years, OT and ML have become increasingly intertwined. This talk contributes to this booming intersection by providing efficient and scalable computational methods for OT and ML.
The first part of the talk shows how neural networks can be used to efficiently approximate the optimal transport map between two densities in high dimensions. To avoid the curse-of-dimensionality, we combine Lagrangian and Eulerian viewpoints and employ neural networks to solve the underlying Hamilton-Jacobi-Bellman equation. Our approach avoids any space discretization and can be implemented in existing machine learning frameworks. We present numerical results for OT in up to 100 dimensions and validate our solver in a two-dimensional setting.
The second part of the talk shows how optimal transport theory can improve the efficiency of training generative models and density estimators, which are critical in machine learning. We consider continuous normalizing flows (CNF) that have emerged as one of the most promising approaches for variational inference in the ML community. Our numerical implementation is a discretize-optimize method whose forward problem relies on manually derived gradients and Laplacian of the neural network and uses automatic differentiation in the optimization. In common benchmark challenges, our method outperforms state-of-the-art CNF approaches by reducing the network size by 8x, accelerate the training by 10x- 40x and allow 30x-50x faster inference.
Francis Bach: On the convergence of gradient descent for wide two-layer neural networks
Abstract: Many supervised learning methods are naturally cast as optimization problems. For prediction models which are linear in their parameters, this often leads to convex problems for which many guarantees exist. Models which are non-linear in their parameters such as neural networks lead to non-convex optimization problems for which guarantees are harder to obtain. In this talk, I will consider two-layer neural networks with homogeneous activation functions where the number of hidden neurons tends to infinity, and show how qualitative convergence guarantees may be derived. I will also highlight open problems related to the quantitative behavior of gradient descent for such models.